Diffraction of limit periodic point sets

Abstract

Limit periodic point sets are aperiodic structures with pure point diffraction supported on a countably, but not finitely generated Fourier module that is based on a lattice and certain integer multiples of it. Examples are cut and project sets with p-adic internal spaces. We illustrate this by explicit results for the diffraction measures of two examples with 2-adic internal spaces. The first and well-known example is the period doubling sequence in one dimension, which is based on the period doubling substitution rule. The second example is a weighted planar point set that is derived from the classic chair tiling in the plane. It can be described as a fixed point of a block substitution rule.

Keywords:

• diffraction
• autocorrelation
• limit periodicity
• substitution systems
• integer inflation factors
• pure point measures

Acknowledgements

This work was supported by the German Research Council (DFG), within the CRC 701, and by a Leverhulme Trust Visiting Professorship Grant (MB).